Typically, vocabulary instruction is associated as part of reading instruction. However, vocabulary instruction is also a necessary component of mathematics instruction. In fact, state and national standards (e.g., Common Core State Standards) require elementary students to communicate about mathematics to provide clear explanations, construct…

To design and improve instruction in mathematical proof, mathematics educators require an adequate definition of proof that is faithful to mathematical practice and relevant to pedagogical situations. In both mathematics education and the philosophy of mathematics, mathematical proof is typically defined as a type of justification that satisfies a…

This article discusses how technology, in conjunction with carefully designed tasks and orchestrated discussions, has the potential to both reveal students' mathematical practices and to provide opportunities to shape those practices. Four specific mathematical practices that tend to be associated with problem-solving situations are examined: (1)…

This article focuses on exploring students' understanding of the concept of function concerning three main aspects: secondary students' ability to (1) define the concept of function and present examples of functions, (2) solve tasks which asked them to recognize and interpret the concept of function presented in different forms of representation,…

Background/Context: Which ideas should be included in the K-12 curriculum, how they are learned, and how they should be taught have been debated for decades in multiple subjects. In this article, we offer mathematics as a case in point of how new standards-related policies may offer an opportunity for reassessment and clarification of such…

The purpose of this article was to describe the task design and implementation of cultural artefacts in a mathematics lesson based on the integration of modelling and conjecturing perspectives. The conceived process of integrating a soccer ball into mathematics lessons via modelling- and conjecturing-based instruction was first detailed. Next, the…

The purpose of this dissertation is to demonstrate one iterate of a process for developing a measurement instrument for student knowledge within educational interventions. Student mathematical knowledge is framed within Cognitively Guided Instruction (CGI) and its tenets. That is, the construct underlying the measurement instrument corresponded…

In this paper we describe an ontological and semiotic model for mathematical knowledge, using elementary combinatorics as an example. We then apply this model to analyze the solving process of some combinatorial problems by students with high mathematical training, and show its utility in providing a semiotic explanation for the difficulty of…

This book discusses the act of understanding in mathematics. The first chapter, "Understanding and Meaning," inquires into the various senses and uses of the word "understanding" in ordinary language and discusses the notion of meaning and relations between understanding and meaning. The second chapter, "Components and…

National Numeracy Strategy (NNS) guidance appears to characterise mathematical language as a set of specialist words with unambiguous definitions, yet analysis of the classroom transcript suggests that at least some mathematical concepts cannot be captured by such definitions. This paper explores the notion of definition within mathematics,…

This document provides a framework for academic and career/technical teachers who want to raise students' achievement through real-life projects reflecting the rigorous academic and technical standards needed for further learning in the workplace and postsecondary education. The document begins with definitions of the six A's of project-based…